Number average molar mass measurement

In the case of a polydisperse polymer it is still the total number n of solute molecules that is measured and the total mass m of solute molecules that is known from sample preparation, resulting in the number average molar mass M = ... 

Mass spectrometry can be used to measure the molar mass distribution (MMD) of a polymer sample by simply measuring the intensity, Nt, of each mass spectral peak with mass m . This is due to the fact that mass spectrometers are equipped with a detector that gives the same response if an ion with mass 1 kDa or 100 Da (actually any mass) strikes against it. In other words, the detector measures the number fraction and this implies that Nt also represents the number of chains with mass m,. Thus, the number-average molar mass, Mn, is given by ... 

Table 13 Number of incorporated monomer units into the 30 triblock copoly(2-oxazoline)s resulting from combined H NMR spectroscopy analyses (top) of the model [A and AB (block co) polymers] and final polymers as well as the measured number average molar masses (Mn,SEC/PDI bottom). H NMR spectra were recorded in CDCI3 or CD2CI2 (PhOx containing polymers) and GPC analyses were performed using DMF (with 5 mM NH4PF6) as eluent. Mn,GPC was calculated utilizing poly(methyl methacrylate) (PMMA) standards...

Measurements of osmotic pressure provide an absolute determination (without calibration) of the number-average molar mass. This is independent of the type of solvent for each solvent the extrapolation to zero concentration results in the same value H/c = RT/Mn. With viscosimetry the determination of M is not absolute dependent on the solvent and the temperature one finds a valne for the intrinsic viscosity, [rj], which is not unique but which has to be calibrated. 

Contrary to viscosity measurements GPC provides number average molar mass data (Mn). For a few studies on the polymerization of BD with Nd-carboxylate-based catalyst systems GPC was systematically applied for the monitoring of Mn as a function of monomer conversion. In these studies three catalyst systems were used (1) NdV/DIBAH/EASC [178], (2) NdV/TIBA/EASC [179] and (3) NdO/TIBA/DEAC [188]. In the first two studies linear Mn-conversion plots were obtained at various molar ratios Al/ Ndv- In these studies, however, molar mass data at low monomer conversions (<20%) are lacking and positive intercepts on the Mn-axis were found. For the ternary catalyst system NdO/TIBA/DEAC used in the third of these studies the concentrations of Nd and TIBA were varied. A linear increase of Mw and Mn on monomer conversion was found. Deviations from linearity were also observed for low monomer conversions. 

The osmotic pressure measurement leads to number-average molar mass since this method is also a means for counting the number of molecules as each molecule contributes equivalently to the measured effect. This can be shown mathematically by remembering that the osmotic pressure n consists additively of... 

In order to obtain the number-average molar mass of a particular sample, osmotic coefficient (II/c) data, measured at various low concentrations, must be extrapolated to the zero concentration limit. In addition to the ideal gas contribution [Eq. (1.73)] that arises from individual polymers, the osmotic pressure also has a contribution from polymer-polymer interactions. The contribution to osmotic pressure from interaction... 

This procedure is analogous to determination of the number-average molar mass and weight-average second virial coefficient from the measurements of osmotic pressure II at different concentrations [Eq. (1.76) and Fig. 1.22]. 

The measurements are often complicated by diffusion of the lower molar mass species through the membrane. As a consequence, the values of M obtained by osmometry may be substantially higher than those measured by other methods. We can show that the value of M obtained is the number-average molar mass, (See Section 35.5 and Problem... 

Already before reporting this combined inifer and living polymerization approach, Kennedy and coworkers developed a controlled isobutene polymerization method based on cumyl ester initiators (Scheme 8.6) with boron trichloride as activator and incremental monomer addition [28], The livingness of the polymerization was demonstrated by the linear increase of number-average molar mass and the constant number of polymer chains (A) with the amount of PIB obtained (wp, as measure for conversion) as well as the narrowing of the molar mass distribution with conversion (Fig. 8.1) [28]. 

Hence, any colligative method should yield the number average molar mass M of a polydisperse polymer. Polymer solutions do not behave in an ideal manner, and nonideal behavior can be eliminated by extrapolating the experimental (F/c) data to c = 0. For example, in the case of boiling point elevation measurements (ebullio-scopy) Equation 9.2 takes the form... 

The sum 2, (c,/ in Equation (9-8) is also contained in the definition of the number-average molar mass = 2,c,7S,[c,7(Af,) ] given in equation (8-44). If this expression is inserted into Equation (9-8), it is seen that osmotic pressure measurements give the number-average molar mass ... 

According to the derivation, Equation (9-23) only applies to solutions at infinite dilution. For finite concentrations, one can, in analogy to the procedure adopted for membrane osmometry measurements, develop a series with virial coefficients. In polymeric solutes, the number-average molar mass is measured in ebulliometry. (The proof is analogous to that given for osmotic-pressure measurements.)... 

Vapor pressure osmometric (thermoelectric, vaporometric) measurements depend on the following principle A drop of a solution with a nonvolatile solute resides on a temperature sensor, i.e., a thermistor. The surrounding region is saturated with solvent vapor. Initially, the drop and vapor are at the same temperature. Since the vapor pressure of the solution is lower than that of the pure solvent, solvent vapor condenses on the solution drop. Because heat of condensation is released, the temperature of the drop rises until the difference in temperature A Tth between the drop and the solvent vapor again eliminates the difference in vapor pressure, so that the chemical potential of the solvent in both phases is equal. An analogous equation to that which applies in ebulliometry is applicable in this case to the relationship between the temperature difference A Tth and the number-average molar mass Mn)= Mioi the solute. 

Figue 11-23. Tensile stress at break (tensile strength) ob of at-poly(styrenes) of narrow molar mass distribution and various number-average molar masses M . Measurements were made at 23°C and 50% relative humidity. Processing was by compression molding ( ) or injection molding (O ) Injection-molded samples are oriented. (After H. W. McCormick, F. M. Brower, and L. Kin.)... 

In this expression, Nj is the number of molecules of molar mass Mi and is the total number of molecules. Measurements of the osmotic pressures of macromolecular solutions yield the number average molar mass. 

For n = 0, Eq. (10.5) reflects the normalization condition. The first moment (n = 1) yields an average value of the distribution function. The second moment (n = 2) contains information about the broadness of the distribution function. The third moment (n = 3) describes the asymmetry of the distribution function. Unfortunately, these moments cannot directly be measured. However, they can be related to quantities accessible by experimental observations. For the characterization of polymers, several average values with respect to molar mass were applied (i.e., number-average molar mass (M ), mass-average molar mass (Mm)-and z-average molar mass (Mz). In polymer thermodynamics, mostly the concept... 

Using the distribution function (i.e., Eq. (10.3)), several average values can be defined. The number-average segment number, which is proportional to the number-average molar mass, which can be measured using the vapor pressure of solutions or osmometry, is given by the ratio of the 0 and -1 moment of the distribution function ... 

The vapor pressure of polyisobutylene (PIB) with a number-average molar mass of JVfpie = 45 kg/mol dissolved in benzene (B) (Mfl = 78.11 g/mol) was measured as function of polymer concentration at 313.15 K. The vapor pressure of benzene at this temperature is 24.3 kPa. For a polymer segment fraction of 0.4 the vapor pressure was 22.6kPa. 

In general a correction of the [/i]-M-relationship is not necessary for a heterogeneity class of A, since the error in this case is smaller than the general error of a viscosimetric measurement (for number average molar masses even [q]-M-rela-tionships of a heterogeneity class B do not need to be corrected). For all lower heterogeneity classes the correction must be done. 

MMD obtained by MS are often compared with averages obtained by SEC. SEC apparatuses found in polymer characterization laboratories are usually equipped with a set of columns that are, for obvious reasons, general purpose. Pores can be all identical or they can be mixed. Modern SEC apparatuses are interfaced with a computer. Powerful software programs perform the data acquisition and data processing. The refractive index (RI) detector measures the weight fraction of chains with mass n. Thus, the number average molar mass, Ma, is given by ... 

Because the number-average molar mass is a colligative property, it can be determined experimentally by techniques which are able to count the number of macromolecules such as osmotic pressure measurements. 

The number average molar mass, Mn, and the polydispersity index, Mw/Mn, for the Pis (Table 1) were determined by gel permeation chromatography (GPQ measurements in N,N-dimethylformamide (DMF), on a PL EMD-950 evaporative tight scattering detector apparatus. Considering that the polymerization degrees of the PAAs and of the... 

Table 4 Experimental number average molar masses for humic acid and Na-humate, measured by membrane osmometry under different solution conditions...

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